Endotracheal intubation is a critical yet technically demanding procedure, with failure or improper tube placement leading to severe complications. Existing robotic and teleoperated intubation systems primarily focus on airway navigation and do not provide integrated control of endotracheal tube advancement or objective verification of tube depth relative to the carina. This paper presents the Robotic Intubation System (BRIS), a compact, human-in-the-loop platform designed to assist fiberoptic-guided intubation while enabling real-time, objective depth awareness. BRIS integrates a four-way steerable fiberoptic bronchoscope, an independent endotracheal tube advancement mechanism, and a camera-augmented mouthpiece compatible with standard clinical workflows. A learning-enabled closed-loop control framework leverages real-time shape sensing to map joystick inputs to distal bronchoscope tip motion in Cartesian space, providing stable and intuitive teleoperation under tendon nonlinearities and airway contact. Monocular endoscopic depth estimation is used to classify airway regions and provide interpretable, anatomy-aware guidance for safe tube positioning relative to the carina. The system is validated on high-fidelity airway mannequins under standard and difficult airway configurations, demonstrating reliable navigation and controlled tube placement. These results highlight BRIS as a step toward safer, more consistent, and clinically compatible robotic airway management.